Conservative modeling of 3-D electromagnetic fields .2. Biconjugate gradient solution and an accelerator

The preceding paper derives a staggered-grid, finite-difference approximation applicable to electromagnetic induction in the Earth. The staggered-grid, finite-difference approximation results in a linear system of equations <(A)under tilde x> = b, where (A) under tilde is symmetric but not Hermitian. This is solved using the biconjugate gradient method, preconditioned with a modified, partial Cholesky decomposition of (A) under tilde. This method takes advantage of the sparsity of (A) under tilde, and converges much more quickly than methods used previously to solve the 3-D induction problem. When simulating a conductivity model at a number of frequencies, the rate of convergence slows as frequency approaches 0. The convergence rate at low frequencies can be improved by an order of magnitude, by alternating the incomplete Cholesky preconditioned biconjugate gradient method with a procedure designed to make the approximate solutions conserve current.